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Properties

Label 20.0.35583470806...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{34}\cdot 11^{18}$
Root discriminant $534.03$
Ramified primes $2, 5, 11$
Class number $5875599300$ (GRH)
Class group $[410, 14330730]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2519337249, 0, 108823443300, 0, 809109957150, 0, 297653591840, 0, 45938756545, 0, 3781457196, 0, 177930500, 0, 4767400, 0, 67980, 0, 440, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 440*x^18 + 67980*x^16 + 4767400*x^14 + 177930500*x^12 + 3781457196*x^10 + 45938756545*x^8 + 297653591840*x^6 + 809109957150*x^4 + 108823443300*x^2 + 2519337249)
 
gp: K = bnfinit(x^20 + 440*x^18 + 67980*x^16 + 4767400*x^14 + 177930500*x^12 + 3781457196*x^10 + 45938756545*x^8 + 297653591840*x^6 + 809109957150*x^4 + 108823443300*x^2 + 2519337249, 1)
 

Normalized defining polynomial

\( x^{20} + 440 x^{18} + 67980 x^{16} + 4767400 x^{14} + 177930500 x^{12} + 3781457196 x^{10} + 45938756545 x^{8} + 297653591840 x^{6} + 809109957150 x^{4} + 108823443300 x^{2} + 2519337249 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3558347080635028147840000000000000000000000000000000000=2^{40}\cdot 5^{34}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $534.03$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2200=2^{3}\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2200}(1,·)$, $\chi_{2200}(1411,·)$, $\chi_{2200}(1281,·)$, $\chi_{2200}(2121,·)$, $\chi_{2200}(1291,·)$, $\chi_{2200}(909,·)$, $\chi_{2200}(79,·)$, $\chi_{2200}(789,·)$, $\chi_{2200}(919,·)$, $\chi_{2200}(731,·)$, $\chi_{2200}(1571,·)$, $\chi_{2200}(549,·)$, $\chi_{2200}(359,·)$, $\chi_{2200}(1961,·)$, $\chi_{2200}(239,·)$, $\chi_{2200}(1841,·)$, $\chi_{2200}(1651,·)$, $\chi_{2200}(629,·)$, $\chi_{2200}(2199,·)$, $\chi_{2200}(1469,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{8} - \frac{1}{9} a^{4} - \frac{7}{27} a^{2} + \frac{1}{3}$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{891} a^{10} - \frac{1}{27} a^{6} - \frac{5}{81} a^{4} - \frac{8}{27} a^{2} - \frac{1}{3}$, $\frac{1}{11583} a^{11} + \frac{1}{351} a^{9} - \frac{19}{351} a^{7} + \frac{121}{1053} a^{5} + \frac{19}{117} a^{3} + \frac{1}{39} a$, $\frac{1}{34749} a^{12} - \frac{19}{34749} a^{10} + \frac{7}{1053} a^{8} - \frac{74}{3159} a^{6} - \frac{154}{3159} a^{4} + \frac{53}{117} a^{2} + \frac{1}{3}$, $\frac{1}{34749} a^{13} - \frac{1}{34749} a^{11} - \frac{1}{1053} a^{9} - \frac{164}{3159} a^{7} + \frac{386}{3159} a^{5} - \frac{109}{1053} a^{3} + \frac{31}{117} a$, $\frac{1}{104247} a^{14} + \frac{1}{104247} a^{12} - \frac{32}{104247} a^{10} - \frac{122}{9477} a^{8} + \frac{472}{9477} a^{6} + \frac{574}{9477} a^{4} - \frac{239}{1053} a^{2} + \frac{4}{9}$, $\frac{1}{104247} a^{15} + \frac{1}{104247} a^{13} + \frac{4}{104247} a^{11} - \frac{14}{9477} a^{9} + \frac{526}{9477} a^{7} + \frac{718}{9477} a^{5} - \frac{23}{1053} a^{3} + \frac{25}{117} a$, $\frac{1}{12196899} a^{16} - \frac{1}{369603} a^{14} - \frac{2}{369603} a^{12} + \frac{2905}{12196899} a^{10} + \frac{5401}{369603} a^{8} + \frac{107}{369603} a^{6} + \frac{114170}{1108809} a^{4} - \frac{41833}{123201} a^{2} - \frac{34}{81}$, $\frac{1}{475679061} a^{17} - \frac{245}{158559687} a^{15} - \frac{256}{158559687} a^{13} - \frac{2009}{475679061} a^{11} - \frac{7118}{14414517} a^{9} - \frac{698968}{14414517} a^{7} + \frac{6500966}{43243551} a^{5} + \frac{417977}{4804839} a^{3} - \frac{1411}{3159} a$, $\frac{1}{200405402331726544657337577087} a^{18} + \frac{48967665281885793952}{66801800777242181552445859029} a^{16} + \frac{21084143954854109279648}{66801800777242181552445859029} a^{14} - \frac{2603816718482912094116507}{200405402331726544657337577087} a^{12} - \frac{30059879575540108965404044}{66801800777242181552445859029} a^{10} + \frac{103046469636602035480989185}{6072890979749289232040532639} a^{8} + \frac{443277533593043286094445999}{18218672939247867696121597917} a^{6} - \frac{252824760428264839462265747}{2024296993249763077346844213} a^{4} + \frac{5115791118400115259955570}{17301683702989428011511489} a^{2} - \frac{4574279312271313257760}{11375202960545317561809}$, $\frac{1}{2204459425648991991230713347957} a^{19} - \frac{178146915538363092589}{200405402331726544657337577087} a^{17} + \frac{24844349652980680508156}{7422422308582464616938428781} a^{15} - \frac{246464046117254015188802}{18218672939247867696121597917} a^{13} + \frac{575273135859954620768681}{200405402331726544657337577087} a^{11} - \frac{71512045534536423802012006}{22267266925747393850815286343} a^{9} - \frac{111270876797898509635063466}{18218672939247867696121597917} a^{7} - \frac{375428435865168179337538154}{18218672939247867696121597917} a^{5} + \frac{331763460005612302909634440}{2024296993249763077346844213} a^{3} - \frac{34903873685620570964897}{1330898746383802154731653} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{410}\times C_{14330730}$, which has order $5875599300$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3112036931534.2773 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{-2}, \sqrt{55})\), 5.5.5719140625.2, 10.0.1071794405000000000000000.1, 10.10.1842146633593750000000000.4, 10.0.58948692275000000000000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.1.0.1}{1} }^{20}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
5Data not computed
$11$11.10.9.4$x^{10} - 99$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.4$x^{10} - 99$$10$$1$$9$$C_{10}$$[\ ]_{10}$